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On the well-posedness of the wave map problem in high dimensions. (English) Zbl 1085.58022

Summary: We construct a gauge theoretic change of variables for the wave map from \(\mathbb{R} \times\mathbb{R}^n\) into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation \(-n\geq 4\) – for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and \(n\geq 4\).

MSC:

58J45 Hyperbolic equations on manifolds
58D25 Equations in function spaces; evolution equations
58J32 Boundary value problems on manifolds
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B33 Critical exponents in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
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